Question

If a = 2 + root under 3 then find the value of a- 1/a . Spammers are not allowed .​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

Given,

$a = 2 + \sqrt{3} \\ \frac{1}{a} = \frac{1}{2 + \sqrt{3} }$

Rationalizing the denominator,

$\frac{1}{a} = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} }$

Using identity :

(x + y)(x - y) = x^2 - y^2

$\frac{1}{a} = \frac{2 - \sqrt{3} }{ {2}^{2} - { \sqrt{3} }^{2} } \\ \frac{1}{a} = \frac{2 - \sqrt{3} }{4 - 3} \\ \frac{1}{a} = 2 - \sqrt{3}$

Now,

$a - \frac{1}{a} = 2 + \sqrt{3} - (2 - \ sqrt{3} ) \\ = 2 + \sqrt{3} - 2 + \sqrt{3} \\ = \sqrt{3} + \sqrt{3} \\ = 2 \sqrt{3}$